We consider the system Applicative_first_order_05__11.

  Alphabet:

    !facminus : [a * a] --> a 
    !facplus : [a * a] --> a 
    !factimes : [a * a] --> a 
    0 : [] --> a 
    1 : [] --> a 
    2 : [] --> a 
    D : [a] --> a 
    cons : [c * d] --> d 
    constant : [] --> a 
    div : [a * a] --> a 
    false : [] --> b 
    filter : [c -> b * d] --> d 
    filter2 : [b * c -> b * c * d] --> d 
    ln : [a] --> a 
    map : [c -> c * d] --> d 
    minus : [a] --> a 
    nil : [] --> d 
    pow : [a * a] --> a 
    t : [] --> a 
    true : [] --> b 

  Rules:

    D(t) => 1 
    D(constant) => 0 
    D(!facplus(x, y)) => !facplus(D(x), D(y)) 
    D(!factimes(x, y)) => !facplus(!factimes(y, D(x)), !factimes(x, D(y))) 
    D(!facminus(x, y)) => !facminus(D(x), D(y)) 
    D(minus(x)) => minus(D(x)) 
    D(div(x, y)) => !facminus(div(D(x), y), div(!factimes(x, D(y)), pow(y, 2))) 
    D(ln(x)) => div(D(x), x) 
    D(pow(x, y)) => !facplus(!factimes(!factimes(y, pow(x, !facminus(y, 1))), D(x)), !factimes(!factimes(pow(x, y), ln(x)), D(y))) 
    map(f, nil) => nil 
    map(f, cons(x, y)) => cons(f x, map(f, y)) 
    filter(f, nil) => nil 
    filter(f, cons(x, y)) => filter2(f x, f, x, y) 
    filter2(true, f, x, y) => cons(x, filter(f, y)) 
    filter2(false, f, x, y) => filter(f, y) 

This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0).

We observe that the rules contain a first-order subset:

  D(t) => 1 
  D(constant) => 0 
  D(!facplus(X, Y)) => !facplus(D(X), D(Y)) 
  D(!factimes(X, Y)) => !facplus(!factimes(Y, D(X)), !factimes(X, D(Y))) 
  D(!facminus(X, Y)) => !facminus(D(X), D(Y)) 
  D(minus(X)) => minus(D(X)) 
  D(div(X, Y)) => !facminus(div(D(X), Y), div(!factimes(X, D(Y)), pow(Y, 2))) 
  D(ln(X)) => div(D(X), X) 
  D(pow(X, Y)) => !facplus(!factimes(!factimes(Y, pow(X, !facminus(Y, 1))), D(X)), !factimes(!factimes(pow(X, Y), ln(X)), D(Y))) 

Moreover, the system is orthogonal.  Thus, by [Kop12, Thm. 7.55], we may omit all first-order dependency pairs from the dependency pair problem (DP(R), R) if this first-order part is terminating when seen as a many-sorted first-order TRS.

According to the external first-order termination prover, this system is indeed terminating:

 || proof of resources/system.trs
 || # AProVE Commit ID: d84c10301d352dfd14de2104819581f4682260f5 fuhs 20130616
 || 
 || 
 || Termination w.r.t. Q of the given QTRS could be proven:
 || 
 || (0) QTRS
 || (1) QTRSRRRProof [EQUIVALENT]
 || (2) QTRS
 || (3) QTRSRRRProof [EQUIVALENT]
 || (4) QTRS
 || (5) RisEmptyProof [EQUIVALENT]
 || (6) YES
 || 
 || 
 || ----------------------------------------
 || 
 || (0)
 || Obligation:
 || Q restricted rewrite system:
 || The TRS R consists of the following rules:
 || 
 ||    D(t) -> 1
 ||    D(constant) -> 0
 ||    D(!facplus(%X, %Y)) -> !facplus(D(%X), D(%Y))
 ||    D(!factimes(%X, %Y)) -> !facplus(!factimes(%Y, D(%X)), !factimes(%X, D(%Y)))
 ||    D(!facminus(%X, %Y)) -> !facminus(D(%X), D(%Y))
 ||    D(minus(%X)) -> minus(D(%X))
 ||    D(div(%X, %Y)) -> !facminus(div(D(%X), %Y), div(!factimes(%X, D(%Y)), pow(%Y, 2)))
 ||    D(ln(%X)) -> div(D(%X), %X)
 ||    D(pow(%X, %Y)) -> !facplus(!factimes(!factimes(%Y, pow(%X, !facminus(%Y, 1))), D(%X)), !factimes(!factimes(pow(%X, %Y), ln(%X)), D(%Y)))
 || 
 || Q is empty.
 || 
 || ----------------------------------------
 || 
 || (1) QTRSRRRProof (EQUIVALENT)
 || Used ordering:
 || D/1(YES)
 || t/0)
 || 1/0)
 || constant/0)
 || 0/0)
 || !facplus/2(YES,YES)
 || !factimes/2(YES,YES)
 || !facminus/2(YES,YES)
 || minus/1)YES(
 || div/2(YES,YES)
 || pow/2(YES,YES)
 || 2/0)
 || ln/1(YES)
 || 
 || Quasi precedence:
 || [D_1, ln_1] > 1 > [!facplus_2, !facminus_2]
 || [D_1, ln_1] > 0 > [!facplus_2, !facminus_2]
 || [D_1, ln_1] > div_2 > !factimes_2 > [!facplus_2, !facminus_2]
 || [D_1, ln_1] > pow_2 > !factimes_2 > [!facplus_2, !facminus_2]
 || [D_1, ln_1] > 2 > [!facplus_2, !facminus_2]
 || t > [!facplus_2, !facminus_2]
 || constant > 0 > [!facplus_2, !facminus_2]
 || 
 || 
 || Status:
 || D_1: multiset status
 || t: multiset status
 || 1: multiset status
 || constant: multiset status
 || 0: multiset status
 || !facplus_2: [2,1]
 || !factimes_2: multiset status
 || !facminus_2: multiset status
 || div_2: multiset status
 || pow_2: multiset status
 || 2: multiset status
 || ln_1: multiset status
 || 
 || With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
 || 
 ||    D(t) -> 1
 ||    D(constant) -> 0
 ||    D(!facplus(%X, %Y)) -> !facplus(D(%X), D(%Y))
 ||    D(!factimes(%X, %Y)) -> !facplus(!factimes(%Y, D(%X)), !factimes(%X, D(%Y)))
 ||    D(!facminus(%X, %Y)) -> !facminus(D(%X), D(%Y))
 ||    D(div(%X, %Y)) -> !facminus(div(D(%X), %Y), div(!factimes(%X, D(%Y)), pow(%Y, 2)))
 ||    D(ln(%X)) -> div(D(%X), %X)
 ||    D(pow(%X, %Y)) -> !facplus(!factimes(!factimes(%Y, pow(%X, !facminus(%Y, 1))), D(%X)), !factimes(!factimes(pow(%X, %Y), ln(%X)), D(%Y)))
 || 
 || 
 || 
 || 
 || ----------------------------------------
 || 
 || (2)
 || Obligation:
 || Q restricted rewrite system:
 || The TRS R consists of the following rules:
 || 
 ||    D(minus(%X)) -> minus(D(%X))
 || 
 || Q is empty.
 || 
 || ----------------------------------------
 || 
 || (3) QTRSRRRProof (EQUIVALENT)
 || Used ordering:
 || Polynomial interpretation [POLO]:
 || 
 ||    POL(D(x_1)) = 2 + 2*x_1
 ||    POL(minus(x_1)) = 1 + x_1
 || With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
 || 
 ||    D(minus(%X)) -> minus(D(%X))
 || 
 || 
 || 
 || 
 || ----------------------------------------
 || 
 || (4)
 || Obligation:
 || Q restricted rewrite system:
 || R is empty.
 || Q is empty.
 || 
 || ----------------------------------------
 || 
 || (5) RisEmptyProof (EQUIVALENT)
 || The TRS R is empty. Hence, termination is trivially proven.
 || ----------------------------------------
 || 
 || (6)
 || YES
 || 
We use the dependency pair framework as described in [Kop12, Ch. 6/7], with static dependency pairs (see [KusIsoSakBla09] and the adaptation for AFSMs and accessible arguments in [FuhKop19]).

We thus obtain the following dependency pair problem (P_0, R_0, computable, formative):

  Dependency Pairs P_0:

    0] map#(F, cons(X, Y)) =#> map#(F, Y)   
    1] filter#(F, cons(X, Y)) =#> filter2#(F X, F, X, Y)   
    2] filter2#(true, F, X, Y) =#> filter#(F, Y)   
    3] filter2#(false, F, X, Y) =#> filter#(F, Y)   

  Rules R_0:

    D(t) => 1 
    D(constant) => 0 
    D(!facplus(X, Y)) => !facplus(D(X), D(Y)) 
    D(!factimes(X, Y)) => !facplus(!factimes(Y, D(X)), !factimes(X, D(Y))) 
    D(!facminus(X, Y)) => !facminus(D(X), D(Y)) 
    D(minus(X)) => minus(D(X)) 
    D(div(X, Y)) => !facminus(div(D(X), Y), div(!factimes(X, D(Y)), pow(Y, 2))) 
    D(ln(X)) => div(D(X), X) 
    D(pow(X, Y)) => !facplus(!factimes(!factimes(Y, pow(X, !facminus(Y, 1))), D(X)), !factimes(!factimes(pow(X, Y), ln(X)), D(Y))) 
    map(F, nil) => nil 
    map(F, cons(X, Y)) => cons(F X, map(F, Y)) 
    filter(F, nil) => nil 
    filter(F, cons(X, Y)) => filter2(F X, F, X, Y) 
    filter2(true, F, X, Y) => cons(X, filter(F, Y)) 
    filter2(false, F, X, Y) => filter(F, Y) 

Thus, the original system is terminating if (P_0, R_0, computable, formative) is finite.

We consider the dependency pair problem (P_0, R_0, computable, formative).

We place the elements of P in a dependency graph approximation G (see e.g. [Kop12, Thm. 7.27, 7.29], as follows:

    * 0 : 0 
    * 1 : 2, 3 
    * 2 : 1 
    * 3 : 1 

This graph has the following strongly connected components:

  P_1:

    map#(F, cons(X, Y)) =#> map#(F, Y)   

  P_2:

    filter#(F, cons(X, Y)) =#> filter2#(F X, F, X, Y)   
    filter2#(true, F, X, Y) =#> filter#(F, Y)   
    filter2#(false, F, X, Y) =#> filter#(F, Y)   

By [Kop12, Thm. 7.31], we may replace any dependency pair problem (P_0, R_0, m, f) by (P_1, R_0, m, f) and (P_2, R_0, m, f).

Thus, the original system is terminating if each of (P_1, R_0, computable, formative) and (P_2, R_0, computable, formative) is finite.

We consider the dependency pair problem (P_2, R_0, computable, formative).

We apply the subterm criterion with the following projection function:

  nu(filter2#) = 4 
  nu(filter#) = 2 

Thus, we can orient the dependency pairs as follows:

  nu(filter#(F, cons(X, Y))) = cons(X, Y) |> Y = nu(filter2#(F X, F, X, Y)) 
  nu(filter2#(true, F, X, Y)) = Y = Y = nu(filter#(F, Y)) 
  nu(filter2#(false, F, X, Y)) = Y = Y = nu(filter#(F, Y)) 

By [FuhKop19, Thm. 61], we may replace a dependency pair problem (P_2, R_0, computable, f) by (P_3, R_0, computable, f), where P_3 contains:

  filter2#(true, F, X, Y) =#> filter#(F, Y)   
  filter2#(false, F, X, Y) =#> filter#(F, Y)   

Thus, the original system is terminating if each of (P_1, R_0, computable, formative) and (P_3, R_0, computable, formative) is finite.

We consider the dependency pair problem (P_3, R_0, computable, formative).

We place the elements of P in a dependency graph approximation G (see e.g. [Kop12, Thm. 7.27, 7.29], as follows:

    * 0 :  
    * 1 :  

This graph has no strongly connected components.  By [Kop12, Thm. 7.31], this implies finiteness of the dependency pair problem.

Thus, the original system is terminating if (P_1, R_0, computable, formative) is finite.

We consider the dependency pair problem (P_1, R_0, computable, formative).

We apply the subterm criterion with the following projection function:

  nu(map#) = 2 

Thus, we can orient the dependency pairs as follows:

  nu(map#(F, cons(X, Y))) = cons(X, Y) |> Y = nu(map#(F, Y)) 

By [FuhKop19, Thm. 61], we may replace a dependency pair problem (P_1, R_0, computable, f) by ({}, R_0, computable, f).  By the empty set processor [Kop12, Thm. 7.15] this problem may be immediately removed.

As all dependency pair problems were succesfully simplified with sound (and complete) processors until nothing remained, we conclude termination.


+++ Citations +++

[FuhKop19]  C. Fuhs, and C. Kop.  A static higher-order dependency pair framework.  In Proceedings of ESOP 2019, 2019.
[Kop12]  C. Kop.  Higher Order Termination.  PhD Thesis, 2012.
[KusIsoSakBla09]  K. Kusakari, Y. Isogai, M. Sakai, and F. Blanqui.  Static Dependency Pair Method Based On Strong Computability for Higher-Order Rewrite Systems.  In volume 92(10) of IEICE Transactions on Information and Systems.  2007--2015, 2009.